SIGACT News Complexity Theory Column 124 Meta-Mathematics of Computational Complexity Theory
This paper addresses fundamental open problems in computational complexity—such as P vs NP and the nonexistence of polynomial-size circuits for SAT—within weak formal systems like bounded arithmetic $S^1_2$. Using a synthesis of proof complexity, model theory, recursion theory, and propositional logic simulation techniques, it establishes, for the first time, rigorous unprovability results for key complexity-theoretic statements in subexponential-strength arithmetic theories. The main contributions are: (1) proving that assertions such as “SAT has no polynomial-size circuits” are independent of $S^1_2$; (2) establishing a tight correspondence between proof complexity lower bounds and circuit lower bounds; and (3) exposing deep metatheoretic barriers preventing any feasible formal proof of P = NP, thereby offering a novel logical foundation for complexity theory.